Group #28335606 — subgroups

Subgroups — #28335606

Order 13696.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293006	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292917	C₂	1	2	yes	maximal, complemented
3	#20292376	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₀₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#555×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133808	—	2	2	no	—
7	#16812109	C₂²	1	1	yes	characteristic
8	#16812040	—	2	2	no	—
9	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
10	#16812030	—	2	2	no	—
11	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	#124×C₁₀₇	—	2	2	no	—
14	#121×C₁₀₇	—	2	2	no	—
15	C₄×D₄×C₁₀₇	C₂²	1	2	yes	—
16	#117×C₁₀₇	—	2	2	no	—
17	C₂×Dic₂₁₄	—	2	1	no	—
18	#13133808	—	4	4	no	—
19	C₂×C₁₀₇⋊C₈	—	2	1	no	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₇	—	2	2	no	—
22	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
23	C₂×D₄×C₁₀₇	—	2	2	no	—
24	C₂×D₄×C₁₀₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	—	2	2	no	—
26	C₄⋊C₄×C₁₀₇	—	2	2	no	—
27	C₄⋊C₄×C₁₀₇	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
29	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄²×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂³×C₁₀₇	—	2	2	no	—
33	Q₈×C₁₀₇	—	4	2	no	—
34	D₄×C₁₀₇	—	2	2	no	—
35	D₄×C₁₀₇	—	2	2	no	—
36	D₄×C₁₀₇	—	8	2	no	—
37	C₂×C₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄×C₁₀₇	—	2	2	no	—
39	C₂×C₄×C₁₀₇	—	4	2	no	—
40	C₂×C₄×C₁₀₇	—	4	2	no	—
41	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
43	C₂×Dic₁₀₇	—	4	1	no	—
44	Dic₂₁₄	—	4	2	no	—
45	C₁₀₇⋊C₈	—	2	2	no	—
46	C₁₀₇⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₇	—	4	2	no	—
48	C₂²×C₁₀₇	—	4	2	no	—
49	C₂²×C₁₀₇	—	4	2	no	—
50	C₂²×C₁₀₇	#120	1	1	yes	characteristic
51	C₄×C₁₀₇	—	2	1	no	—
52	C₄×C₁₀₇	—	4	2	no	—
53	C₄×C₁₀₇	—	4	2	no	—
54	C₄×C₁₀₇	#135	1	2	yes	—
55	C₄×C₁₀₇	#137	1	2	yes	—
56	Dic₁₀₇	—	8	1	no	—
57	C₂×C₁₀₇	—	8	2	no	—
58	C₂×C₁₀₇	#452	1	2	yes	—
59	C₂×C₁₀₇	#453	1	1	yes	characteristic
60	#1634	—	107	1	no	maximal, Sylow
61	C₁₀₇	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	D₁₀₇	1	1	yes	characteristic
63	#494	—	107	1	no	—
64	#476	—	107	2	no	—
65	#333	—	107	1	no	—
66	D₄⋊C₈	—	107	2	no	—
67	C₂×Q₁₆	—	214	2	no	—
68	C₄⋊Q₈	—	107	1	no	—
69	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₀₇	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	214	2	no	—
75	C₄⋊C₈	—	107	2	no	—
76	#102	—	214	2	no	—
77	C₄×C₈	—	107	1	no	—
78	C₂×Q₈	—	214	1	no	—
79	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₀₇	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	428	4	no	—
84	C₂×C₈	—	214	2	no	—
85	C₂×C₈	—	214	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	214	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₀₇	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	428	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	428	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
106	C₈	—	428	2	no	—
107	C₈	—	214	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#16812078	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	856	1	no	—
116	C₄	#16812129	1	2	yes	—
117	C₄	#16812139	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#20292789	1	2	yes	—
120	C₂	#20292795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293006

C₂

yes

maximal, characteristic, complemented

#20292917

C₂

yes

maximal, complemented

#20292376

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, complemented

#555×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133808

—

#16812109

C₂²

yes

characteristic

#16812040

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812030

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

#124×C₁₀₇

—

#121×C₁₀₇

—

C₄×D₄×C₁₀₇

C₂²

yes

—

#117×C₁₀₇

—

C₂×Dic₂₁₄

—

#13133808

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₂²⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#120

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#135

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#452

yes

—

C₂×C₁₀₇

#453

yes

characteristic

#1634

—

107

maximal, Sylow

C₁₀₇

#1634

yes

characteristic, complemented, Sylow

#555

D₁₀₇

yes

characteristic

#494

—

107

—

#476

—

107

—

#333

—

107

—

D₄⋊C₈

—

107

—

C₂×Q₁₆

—

214

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₀₇

yes

—

#117

—

#106

—

214

—

C₄⋊C₈

—

107

—

#102

—

214

—

C₄×C₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

—

Q₁₆

—

428

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

C₂²⋊D₁₀₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

428

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

428

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

106

C₈

—

428

—

107

C₈

—

214

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#16812078

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

856

—

116

C₄

#16812129

yes

—

117

C₄

#16812139

yes

—

118

C₂

—

119

C₂

#20292789

yes

—

120

C₂

#20292795

yes

characteristic

	subgroup	G/H
center	C₂²	#16812078
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting, prime radical	#555×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#120
hypercenter	#555	D₁₀₇
hypocenter	C₁₀₇	#1634
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#16812078

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting, prime radical

#555×C₁₀₇

C₂

C₂²×C₁₀₇

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #20293006	C₂	semidirect
2. #20292917	C₂	semidirect
3. #20292376	C₂	semidirect
4. D₄⋊(C₁₀₇⋊C₈)	C₂	semidirect
61. C₁₀₇	#1634	semidirect

H ≅ G/N

type

1. #20293006

C₂

semidirect

2. #20292917

C₂

semidirect

3. #20292376

C₂

semidirect

4. D₄⋊(C₁₀₇⋊C₈)

C₂

semidirect

61. C₁₀₇

#1634

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20293006	2
#20292917	2
#20292376	2
D₄⋊(C₁₀₇⋊C₈)	2
#555×C₁₀₇	2
#1634	107

subgroup

index

#20293006

#20292917

#20292376

D₄⋊(C₁₀₇⋊C₈)

#555×C₁₀₇

#1634

107

Subgroups — #28335606

Conjugacy classes of subgroups

Complements

Maximal subgroups